Borderline Weak--Type Estimates for Sparse Bilinear Forms Involving $A_{\infty }$ Maximal Functions

Abstract

For any operator $T$ whose bilinear form can be dominated by a sparse bilinear form, we prove that $T$ is bounded as a map from $L^1(\tilde{M}w)$ into weak--$L^1(w)$. Our main innovation is that $\tilde{M}$ is a maximal function defined by directly using the local $A_{\infty }$ characteristic of the weight (rather than Orlicz norms). Prior results are due to Coifman\&Fefferman, Pérez, Hytönen\&Pérez, and Domingo-Salazar\&Lacey\&Rey. As we discuss, but do not prove, the maximal functions we use seem to be on the order of $M_{L(loglogL)(logloglogL)(loglogloglogL{)}^{1+\epsilon }}$.

Updated based on referee's comments - see paper for acknowledgement

References in corpus (1)

• A sparse domination principle for rough singular integrals